American Institute of Mathematical Sciences

Advanced
November  2019, 12(7): 2163-2175. doi: 10.3934/dcdss.2019139

Ground state homoclinic solutions for a second-order Hamiltonian system

Xiaoping Wang

Department of Mathematics, Xiangnan University, Chenzhou, Hunan 42300, China

Received  January 2018 Revised  May 2018 Published  December 2018

Fund Project: This work is partially supported by the NNFC (No: 11471278) of China and Key project of Hunan Provincial Education Department (13A093).

Consider the second-order Hamiltonian system
$ \ddot{u}-L(t)u+\nabla W(t, u) = 0, $
where
$ t\in {\mathbb{R}}, u\in {\mathbb{R}}^{N} $
,
$ L: \mathbb{R}\rightarrow {\mathbb{R}}^{N\times N} $
 and
$ W: {\mathbb{R}}\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}} $
. We mainly study the case when both
$ L $
 and
$ W $
 are periodic in
$ t $
 and
$ 0 $
 belongs to a spectral gap of
$ \sigma\left(-\frac{d^2}{dt^2} +L\right) $
. We prove that the above system possesses a ground state homoclinic solution under assumptions which are weaker than the ones known in the literature.
Keywords: Homoclinic solution, hamiltonian system, periodic potentials, ground state homoclinic solution, variational method.
Mathematics Subject Classification: Primary: 34C37; Secondary: 58E05.
Citation: Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139
References:
[1]

A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, Rend. Semmin. Mat. Univ. Padova, 89 (1993), 177-194.   Google Scholar

[2]

F. Antonacci and P. Magrone, Second order nonautonomous systems with symmetric potential changing sign, Rend. Mat. Appl., 18 (1998), 367-379.   Google Scholar

[3]

P. Caldiroli and P. Montechiari, Homoclinic orbites for second order Hamiltonian systems with potential changing sign, J. Commu. Appl. Nonlinear Anal., 1 (1994), 97-129.   Google Scholar

[4]

S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin Dyn Syst-A, 38 (2018), 2333-2348.  doi: 10.3934/dcds.2018096.  Google Scholar

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V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.  doi: 10.1007/BF01444526.  Google Scholar

[6]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.  doi: 10.1090/S0894-0347-1991-1119200-3.  Google Scholar

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Y. H. Ding and M. Girardi, Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign, Dynam. Systems Appl., 2 (1993), 131-145.   Google Scholar

[8]

Y. H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113.  doi: 10.1016/0362-546X(94)00229-B.  Google Scholar

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Y. H. Ding and C. Lee, Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413.  doi: 10.1016/j.na.2008.10.116.  Google Scholar

[10] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.   Google Scholar
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Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-9029-8.  Google Scholar

[12]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472.   Google Scholar

[13]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.  Google Scholar

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Y. Lv and C. L. Tang, Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198.  doi: 10.1016/j.na.2006.08.043.  Google Scholar

[15]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential Equations, 219 (2003), 375-389.  doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[16]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.   Google Scholar

[17]

Z. Q. Ou and C. L. Tang, Existence of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213.   Google Scholar

[18]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[19]

P. H. Rabinowitz, Periodic and Heteroclinic orbits for a periodic Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 331-346.  doi: 10.1016/S0294-1449(16)30314-6.  Google Scholar

[20]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  doi: 10.1017/S0308210500024240.  Google Scholar

[21]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.  doi: 10.1007/BF02571356.  Google Scholar

[22]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[23]

X. H. Tang, Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on $ \mathbb{R} ^N$, Canad. Math. Bull., 58 (2015), 651-663.  doi: 10.4153/CMB-2015-019-2.  Google Scholar

[24]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.  Google Scholar

[25]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ, (2018), 1-15. https://doi.org/10.1007/s10884-018-9662-2. Google Scholar

[26]

X. H. Tang and L. Xiao, Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl., 351 (2009), 586-594.  doi: 10.1016/j.jmaa.2008.10.038.  Google Scholar

[27]

X. H. Tang and L. Xiao, Homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 71 (2009), 1140-1152.  doi: 10.1016/j.na.2008.11.038.  Google Scholar

[28]

X. H. Tang and X. Y. Lin, Homoclinic solutions for a class of second-order Hamiltonian systems, J. Math. Anal. Appl., 354 (2009), 539-549.  doi: 10.1016/j.jmaa.2008.12.052.  Google Scholar

[29]

X. H. Tang and X. Y. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. Royal Soc. Edinburgh A, 141 (2011), 1103-1119.  doi: 10.1017/S0308210509001346.  Google Scholar

[30]

J. Wang, F. Zhang and J. Xu, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 366 (2010), 569-581.  doi: 10.1016/j.jmaa.2010.01.060.  Google Scholar

[31]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[32]

D. L. Wu, X. P. Wu and C. L. Tang, Homoclinic solutions for a class of nonperiodic and noneven second-order Hamiltonian systems, J. Math. Anal. Appl., 367 (2010), 154-166.  doi: 10.1016/j.jmaa.2009.12.046.  Google Scholar

[33]

D. L. Wu, X. P. Wu and C. L. Tang, Subharmonic and homoclinic solutions for second order Hamiltonian systems with new superquadratic conditions, Chaos Solitons and Fractals, 73 (2015), 183-190.  doi: 10.1016/j.chaos.2015.01.019.  Google Scholar

[34]

X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398.  doi: 10.1016/j.na.2011.03.059.  Google Scholar

[35]

J. Yang and F. Zhang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic potentials, Nonlinear Anal.RWA, 10 (2009), 1417-1423.  doi: 10.1016/j.nonrwa.2008.01.013.  Google Scholar

[36]

M. H. Yang and Z. Q. Han, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials, Nonlinear Anal.RWA, 12 (2011), 2742-2751.  doi: 10.1016/j.nonrwa.2011.03.019.  Google Scholar

[37]

Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part, Commun. Pure Appl. Anal., 14 (2015), 1929-1940.  doi: 10.3934/cpaa.2015.14.1929.  Google Scholar

[38]

Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143 (2015), 3155-3163.  doi: 10.1090/S0002-9939-2015-12107-7.  Google Scholar

[39]

Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212.  doi: 10.1016/j.jde.2010.03.010.  Google Scholar

[40]

W. M. Zou, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287.  doi: 10.1016/S0893-9659(03)90130-3.  Google Scholar

show all references

References:
[1]

A. Ambrosetti and V. Coti Zelati, Multiple homoclinic orbits for a class of conservative systems, Rend. Semmin. Mat. Univ. Padova, 89 (1993), 177-194.   Google Scholar

[2]

F. Antonacci and P. Magrone, Second order nonautonomous systems with symmetric potential changing sign, Rend. Mat. Appl., 18 (1998), 367-379.   Google Scholar

[3]

P. Caldiroli and P. Montechiari, Homoclinic orbites for second order Hamiltonian systems with potential changing sign, J. Commu. Appl. Nonlinear Anal., 1 (1994), 97-129.   Google Scholar

[4]

S. T. Chen and X. H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin Dyn Syst-A, 38 (2018), 2333-2348.  doi: 10.3934/dcds.2018096.  Google Scholar

[5]

V. Coti Zelati, I. Ekeland and E. Sere, A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann., 288 (1990), 133-160.  doi: 10.1007/BF01444526.  Google Scholar

[6]

V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727.  doi: 10.1090/S0894-0347-1991-1119200-3.  Google Scholar

[7]

Y. H. Ding and M. Girardi, Periodic and homoclinic solutions to a class of Hamiltonian systems with the potentials changing sign, Dynam. Systems Appl., 2 (1993), 131-145.   Google Scholar

[8]

Y. H. Ding, Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems, Nonlinear Anal., 25 (1995), 1095-1113.  doi: 10.1016/0362-546X(94)00229-B.  Google Scholar

[9]

Y. H. Ding and C. Lee, Homoclinics for asymptotically quadratic and superquadratic Hamiltonian systems, Nonlinear Anal., 71 (2009), 1395-1413.  doi: 10.1016/j.na.2008.10.116.  Google Scholar

[10] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987.   Google Scholar
[11]

Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-9029-8.  Google Scholar

[12]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472.   Google Scholar

[13]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.  Google Scholar

[14]

Y. Lv and C. L. Tang, Existence of even homoclinic orbits for second-order Hamiltonian systems, Nonlinear Anal., 67 (2007), 2189-2198.  doi: 10.1016/j.na.2006.08.043.  Google Scholar

[15]

M. Izydorek and J. Janczewska, Homoclinic solutions for a class of second order Hamiltonian systems, J. Differential Equations, 219 (2003), 375-389.  doi: 10.1016/j.jde.2005.06.029.  Google Scholar

[16]

W. Omana and M. Willem, Homoclinic orbits for a class of Hamiltonian systems, Differential Integral Equations, 5 (1992), 1115-1120.   Google Scholar

[17]

Z. Q. Ou and C. L. Tang, Existence of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 291 (2004), 203-213.   Google Scholar

[18]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.  Google Scholar

[19]

P. H. Rabinowitz, Periodic and Heteroclinic orbits for a periodic Hamiltonian systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 6 (1989), 331-346.  doi: 10.1016/S0294-1449(16)30314-6.  Google Scholar

[20]

P. H. Rabinowitz, Homoclinic orbits for a class of Hamiltonian systems, Proc, Roy. Soc. Edinburgh Sect. A, 114 (1990), 33-38.  doi: 10.1017/S0308210500024240.  Google Scholar

[21]

P. H. Rabinowitz and K. Tanaka, Some results on connecting orbits for a class of Hamiltonian systems, Math. Z., 206 (1991), 473-499.  doi: 10.1007/BF02571356.  Google Scholar

[22]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[23]

X. H. Tang, Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on $ \mathbb{R} ^N$, Canad. Math. Bull., 58 (2015), 651-663.  doi: 10.4153/CMB-2015-019-2.  Google Scholar

[24]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.  Google Scholar

[25]

X. H. Tang, X. Y. Lin and J. S. Yu, Nontrivial solutions for Schrödinger equation with local super-quadratic conditions, J. Dyn. Differ. Equ, (2018), 1-15. https://doi.org/10.1007/s10884-018-9662-2. Google Scholar

[26]

X. H. Tang and L. Xiao, Homoclinic solutions for nonautonomous second order Hamiltonian systems with a coercive potential, J. Math. Anal. Appl., 351 (2009), 586-594.  doi: 10.1016/j.jmaa.2008.10.038.  Google Scholar

[27]

X. H. Tang and L. Xiao, Homoclinic solutions for a class of second-order Hamiltonian systems, Nonlinear Anal., 71 (2009), 1140-1152.  doi: 10.1016/j.na.2008.11.038.  Google Scholar

[28]

X. H. Tang and X. Y. Lin, Homoclinic solutions for a class of second-order Hamiltonian systems, J. Math. Anal. Appl., 354 (2009), 539-549.  doi: 10.1016/j.jmaa.2008.12.052.  Google Scholar

[29]

X. H. Tang and X. Y. Lin, Existence of infinitely many homoclinic orbits in Hamiltonian systems, Proc. Royal Soc. Edinburgh A, 141 (2011), 1103-1119.  doi: 10.1017/S0308210509001346.  Google Scholar

[30]

J. Wang, F. Zhang and J. Xu, Existence and multiplicity of homoclinic orbits for the second order Hamiltonian systems, J. Math. Anal. Appl., 366 (2010), 569-581.  doi: 10.1016/j.jmaa.2010.01.060.  Google Scholar

[31]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[32]

D. L. Wu, X. P. Wu and C. L. Tang, Homoclinic solutions for a class of nonperiodic and noneven second-order Hamiltonian systems, J. Math. Anal. Appl., 367 (2010), 154-166.  doi: 10.1016/j.jmaa.2009.12.046.  Google Scholar

[33]

D. L. Wu, X. P. Wu and C. L. Tang, Subharmonic and homoclinic solutions for second order Hamiltonian systems with new superquadratic conditions, Chaos Solitons and Fractals, 73 (2015), 183-190.  doi: 10.1016/j.chaos.2015.01.019.  Google Scholar

[34]

X. Wu and W. Zhang, Existence and multiplicity of homoclinic solutions for a class of damped vibration problems, Nonlinear Anal., 74 (2011), 4392-4398.  doi: 10.1016/j.na.2011.03.059.  Google Scholar

[35]

J. Yang and F. Zhang, Infinitely many homoclinic orbits for the second order Hamiltonian systems with super-quadratic potentials, Nonlinear Anal.RWA, 10 (2009), 1417-1423.  doi: 10.1016/j.nonrwa.2008.01.013.  Google Scholar

[36]

M. H. Yang and Z. Q. Han, The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials, Nonlinear Anal.RWA, 12 (2011), 2742-2751.  doi: 10.1016/j.nonrwa.2011.03.019.  Google Scholar

[37]

Q. Q. Zhang, Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part, Commun. Pure Appl. Anal., 14 (2015), 1929-1940.  doi: 10.3934/cpaa.2015.14.1929.  Google Scholar

[38]

Q. Q. Zhang, Homoclinic orbits for a class of discrete periodic Hamiltonian systems, Proc. Amer. Math. Soc., 143 (2015), 3155-3163.  doi: 10.1090/S0002-9939-2015-12107-7.  Google Scholar

[39]

Z. Zhou and J. S. Yu, On the existence of homoclinic solutions of a class of discrete nonlinear periodic systems, J. Differential Equations, 249 (2010), 1199-1212.  doi: 10.1016/j.jde.2010.03.010.  Google Scholar

[40]

W. M. Zou, Infinitely many homoclinic orbits for the second-order Hamiltonian systems, Appl. Math. Lett., 16 (2003), 1283-1287.  doi: 10.1016/S0893-9659(03)90130-3.  Google Scholar

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